ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 2 Issue 2 (2010), Pages 18-26.
REGULARIZATION FOR A CLASS OF BACKWARD PARABOLIC PROBLEMS
NGUYEN HUY TUAN
Abstract. The backward Cauchy problemut+Au(t) = 0, u(T) =f,whereA is a positive self-adjoint unbounded operator,which has continuous spectrum andf is a given function being given is regularized by the well-posed problem the truncation method. The new error estimates of the regularized solution are obtained. The main purpose of this paper is to improve the earlier results by [2, 19].
1. Introduction
LetH be a Hilbert space. For a positive numberT, we shall consider the problem of finding the functionu: [0, T]→H from the system
ut+Au= 0, 0< t < T
u(T) =f (1.1)
for some prescribed final value f in H. The operator A is a positive self-adjoint operator such that 0∈ρ(A). This problem is well known to be severely ill-posed and regularization methods for it are required.
The case A be a self-adjoint operator having the discrete spectrum on H has been considered by many authors, using different approaches. Such authors as Latt‘es and Lions [9],Miller [10], Payne [12] have approximated (1) by perturbing the operator A. This method is called Quasi-reversibility method (QR). The main ideas of the method is of adding a ”corrector” into the main equation. In fact, they considered the problem
(ut+Au−A∗Au= 0, 0< t < T
u(T) =f (1.2)
The stability magnitude of the method are of order ec−1. In [7], the problem is approximated with
ut+Au+Aut= 0, 0< t < T
u(T) =f (1.3)
2000Mathematics Subject Classification. 35K05, 35K99, 47J06, 47H10.
Key words and phrases. Parabolic problem, Backward problem, semigroup of operator , Ill- posed problem, Contraction principle.
c
2010 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.
Submitted September, 2009. Published April, 2010.
18
Ames and Hughes [1] gave a survey about an association between the operator- theoretic methods and the QR method to treat the abstract Cauchy problem
du
dt =Au, u(T) =χ, 0< t < T.
The authors considered the problem in both the Hilbert space and in the Banach space. They also gave many structural stability results. Very recently, using the QR method, Yongzhong Huang and Quan Zheng, in [17], considered the problem in an abstract setting, i.e.,−Ais the generator of an analytic semigroup in a Banach space.
In [14], Showalter presented a different method called the quasiboundary value (QBV) method to regularize that linear homogeneous problem which gave a sta- bility estimate better than the one of discussed methods. The main ideas of the method is of adding an appropriate ”corrector” into the final data. Using the method, Clark-Oppenheimer, in [3]in regularizing the problem (1.1) by the non- local boundary value problem
ut+Au= 0, 0< t < T
u(0) +u(T) =f (1.4)
Very recently, Denche-Bessila in [18], regularized the backward problem by pertub- ing the final condition as follows
ut+Au= 0, 0< t < T
ut(0) +u(T) =f (1.5)
In our knowledge, the caseAhas discrete spectrum has been treated in many recent papers, such as [16, 18]. However, the literature on the homogeneous case of the problem in the caseAhas continuous spectrum are quite scarce. For some related works on this type of such problem, we refer the reader to N.Boussetila and F.
Rebbani [2], Denche and S. Djezzar [19].
In the present paper, we shall use new truncated method to extend the continuous dependence results of [2, 19]. Recently,the truncated regularization method has been effectively applied to solve the sideways heat equation [4], a more general sideways parabolic equation [5] and backward heat [6]. This regularization method is rather simple and convenient for dealing with some ill-posed problems. However, as far as we know, there are not any results of truncated method for treating the problem (1) until now. Moreover, we establish some new error estimates including the order of Holder type. Especially, the convergence of the approximate solution att= 0 is also proved.
This paper is organized as follows. In the next section, for ease of the reading, we summarize some well-known facts in semigroup of operator. The stability estimates of the regularized solution will be presented in Section 3.
Before going to the details of next sections, we shall give the precise formula of the operator S(t). We assume that H is a separable Hilbert space and A is self-adjoint and that 0 is in the resolvent set ofA. S(t) is the compact contraction semi group generated by−A. We denote by{Eλ, λ≥0}the spectral resolution of the identity associated toA. ThenS(t) =e−tA=R∞
0 e−tλdEλ∈ L(H),t≥0, the
C0-semigroup generated by −A. Then from [13], we get Au=
Z +∞
0
λdEλu (1.6)
for allu∈D(A). In this connection, u∈D(A) iff the integral (1.6) exists,i.e., Z +∞
0
λ2dkEλuk2<∞.
2. The main results
A solution of (1.1) on the interval [0, T] is a functionu∈ C([0, T];H)∩C1((0, T);H) such that for allt ∈(0, T), u(t)∈ D(A) andu(T) =f holds. It is useful to know exactly the admissible set for which (1.1) has a solution. The following lemma gives an answer to this question.
Lemma 2.1. Problem (1.1) has a solution if and only if Z ∞
0
e2λTdkEλfk2<∞ and its unique solution is represented by
u(t) =e(T−t)Af. (2.1)
If the problem (1.1) admits a solutionuthen this solution can be represented by u(t) = e(T−t)Af =
Z ∞
0
eλ(T−t)dEλf. (2.2) Sincet < T, we know from (2.2) that the termse−(t−T)λ is the unstability cause.
So, to regularize problem (2.2),we should replace it by the better terms. In [19], the authors replacede−(t−T)λ by the better term λ+ee−tλ−T λ . In this paper, we hope to recover the stability of problem (2.2) by filtering the high frequencies with suitable method. The essence of our regularization method is just to eliminate all high frequencies from the solution, and instead consider (2.2) only forλ≤β, whereβ is an appropriate positive constant depend onwhich will be selected appropriately as regularization parameter. Let f andf denote the exact and measured data at t = T, respectively, which satisfy
kf−fk ≤.
Hence, the ill-posed problem (1.1) can be approximated by the problem u(t) =
Z ∞
0
eλ(T−t)χ[0,β]dEλf (2.3) whereχ[a,b] is the characteristic function of interval [a, b] fora < b.
The approximated solutionvcorresponding to the final valuefis given the form v(t) =
Z ∞
0
eλ(T−t)χ[0,β]dEλf. (2.4) For clarity, from now on, we denote the solution of (1) byu(t), and the solution of (2.4) byv(t). Our first main theorem is the following
Theorem 2.2. The solution defined in(2.4)depends continuously (inC([0, T];H)) on f. Let v and w be two solutions of problem (2.4)corresponding to the final values f andg respectively, then
kv(t)−w(t)k ≤e(T−t)βkf−gk.
Remark. 1) If β= T1 ln( T
(1+ln(T))), the stability magnitude is E1(, t) =C1e(T−t)β=C1Tt−1 T−1ln(T e−1)Tt−1
. Note that the stability order in [19] is the form E2() =C2 T
(1+ln(T)). Comparing E1(, t)andE2(, t), we see that the stability order of our method is less than its in [19].
2. Using Theorem 2.2, we have the error
ku(t)−v(t)k ≤e(T−t)βkf−fk ≤e(T−t)β. (2.5) Theorem 2.3. Let u∈C([0, T];H)be a solution of (1.1). Then
ku(t)−v(t)k ≤e(T−t)β+e−tβku(0)k, ∀t∈(0, T]. (2.6) Remark. 1. If we chooseβ =T1 ln(1)then the estimates (2.6)becomes
ku(t)−u(t)k ≤Tt(1 +ku(0)k). (2.7) This error is also given by Clark and Oppenheimer[3], Tautenhahn[20].
2. The error in t = 0 is not considered in (2.6). In the next Theorem, we shall establish some estimates which convergences to zero in t= 0.
Theorem 2.4. Assume thatuhas the eigenfunction expansionu(t) =R∞
0 dEλu(t).
a) Assume that there exist some positive constantspandI1 such that Z ∞
0
λ2pdkEλu(t)k2< I12. (2.8) If we chooseβ =Ta ln(1),(0< a <1),then for every t∈[0, T]
ku(t)−v(t)k ≤atT+1−a+ (T a)pI1
ln(1
) −p
. (2.9)
b) Assume that there exist some positive constantsq andI2 such that Z ∞
0
e2qλdkEλu(t)k2< I22. (2.10) If we chooseβ =T1+qln(1),then for every t∈[0, T]
ku(t)−v(t)k ≤Tq+q
T+qt +I2
. (2.11)
Remark. 1. We know that the exact solutionuof (1.1) is unknown. Therefore, in practice, the assumptions a) and b) in Theorem 2.4 are very difficult to check. To improve this, we give the different conditions on the known functionf as follows
Z ∞
0
λ2pe2(T−t)λdkEλfk2< I32. (2.12) and
Z ∞
0
e2(T−t+q)λdkEλfk2< I42. (2.13)
whereI3andI4 are the positive numbers. Then, by similar way, we obtain the same convergence result.
2. One superficial advantage of this method is that there is a error estimation in the original timet= 0, which is not given in[22]. We have the following estimate int= 0
ku(0)−v(0)k ≤1−a+ (T a)pI1
ln(1
) −p
. (2.14)
and
ku(0)−v(0)k ≤T+qq (1 +I2). (2.15) 3. It follows from (2.14) that if →0 then the second term on the righthand side of the inequality approaches zero with a logarithmic speed, and the first one as a power.So, the terms in (2.14) is logarithmic stability estimates. This logarithmic order is also given in[2, 6, 16, 20, 21].
4. Notice that the error is in[19] (See Theorem 2.6 , page 5).
ku(0)−u(0)k ≤ N T ekT
1 + lnT )
−1
. (2.16)
In [2](see Theorem 4.14, p.12), the authors established the optimal order error of their method. With the conditionkAu(0)k2=R+∞
γ λ2e2T λdkEλϕk2<∞, and that γ≥1, they estimatedkuσ(0)−u(0)k2 as follows
kuσ(0)−u(0)k2≤2 T 1 + ln(γTβ )
2 +T α
kAu(0)k2. (2.17) The error orders are same in (2.14).
5. It follows from (2.15)we obtain the Holder stability. As we know, the error of Holder form is the optimal error. Thus,the convergence to zero ofα, (0< α <1)is quickly than logarithmic terms. We note again such order is not considered in[16].
Comparing (2.15) and (2.16) with (2.17) and the results obtained in [16, 18, 19], we realize (2.15) is sharp and the best known estimate. The convergence to zero of α, (0< α <1) is quickly than logarithmic terms. This is generalization of many discussed results.
3. Proof of the main results.
Proof of Lemma 2.1.
If the problem (1.1) has a unique solutionuthen
u(t) =e−tAu(0). (3.1)
Thenu(T) =f =e−tAu(0). It follows that ku(.,0)k2=
Z ∞
0
e2T λdkEλfk2<∞.
If we get (7), then definev be as the function v=
Z ∞
0
eT λdEλf.
Consider the problem
ut+Au= 0,
u(0) =v, (3.2)
It is clear to see that (3.2) is the direct problem so it has a unique solutionu. We have
u(t) = Z ∞
0
e−tλeT λdEλf. (3.3)
Lett=T in (3.3), we have u(T) =
Z ∞
0
e−T λeT λdEλf =f.
Hence,uis the unique solution of (1.1).
Proof of Theorem 2.2.
It is well known that for allt∈[0, T], v(t)−w(t) =
Z β
0
eλ(T−t)dEλ(f−g). (3.4) Using (3.4), we obtain
kv(t)−w(t)k2 ≤ e2(T−t)β Z ∞
0
dkEλ(f−g)k2
≤ e2(T−t)βkf−gk2.
This inequality follows the solution of the problem (2.4) depend continuously onϕ and Theorem 2.2 is proved.
Proof of Theorem 2.3.
The functionsu(t), u(t) have the expansion u(t) =
Z ∞
0
eλ(T−t)dEλf. (3.5)
u(t) = Z ∞
0
eλ(T−t)χ[0,β]dEλf. (3.6) Hence, we get
u(t)−u(t) = Z ∞
β
eλ(T−t)dEλf = Z ∞
0
e−λtχ[β,∞]eλTdEλf.
Then
ku(t)−u(t)k2 ≤ Z ∞
0
e−λtχ[A,∞]2
eλTdEλf2 . Using
e−λtχ[β,∞]2
≤e−2tβ and
ku(0)k2= Z ∞
0
eλTdEλf2 .
we obtain
ku(t)−u(t)k2 ≤ e−2tβku(0)k2.
Using (2.5) and the triangle inequality, we get
ku(t)−v(t)k ≤ ku(t)−u(t)k+ku(t)−v(t)k
≤ e−tβku(0)k+e(T−t)β.
This completes the proof of Theorem 2.3.
Proof of Theorem 2.4
Proof a. Since (3.5) and (3.6), we have u(t)−u(t) =
Z ∞
0
eλ(T−t)χ[β,∞]dEλf
= Z ∞
0
λ−βeλ(T−t)λβχ[β,∞]dEλf.
Then
ku(t)−u(t)k2 = Z ∞
0
λ−βχ[β,∞]2
eλ(T−t)λβdEλf2
≤ β−2p Z ∞
0
λ2pdkEλu(t)k2. Using (2.5) and the triangle inequality, we get
ku(t)−v(t)k ≤ ku(t)−u(t)k+ku(t)−v(t)k ≤ I1β−p+e(T−t)β
≤ atT+1−a+ (T a)pI1
ln(1
) −p
. Proof b. Since (3.5) and (3.6), we also have
u(t)−u(t) = Z ∞
0
eλ(T−t)χ[β,∞]dEλf
= Z ∞
0
e−qλeλ(T−t)eqλχ[β,∞]dEλf.
Then
ku(t)−u(t)k2 = Z ∞
0
e−qλχ[β,∞]2
eλ(T−t)eqλdEλf2
≤ e−2qβ Z ∞
0
e2qλdkEλu(t)k2.
Using (2.5) and the triangle inequality, we get
ku(t)−v(t)k ≤ ku(t)−u(t)k+ku(t)−v(t)k ≤ I2e−2qβ Z ∞
0
e2qλdkEλu(t)k2+e(T−t)β
≤ T+qq
Tt+q +I2
.
Acknowledgments. The author would like to thank the anonymous referees for their valuable comments leading to the improvement of our manuscript.
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Nguyen Huy Tuan
Department of Mathematics, Sai Gon University, 273 An Duong Vuong , Ho Chi Minh city, Vietnam
E-mail address:tuanhuy [email protected]
